The concept of center as an equivariant map and a proof of an analogue of the center conjecture for equifacetal simplices

TL;DR

This paper models geometric centers as equivariant maps between G-spaces and proves an analogue of the center conjecture for equifacetal simplices, extending understanding of symmetry-invariant centers.

Contribution

It introduces a general framework for centers as equivariant maps and proves an analogue of the center conjecture for equifacetal simplices under certain conditions.

Findings

Existence of equivariant maps for given fixed points and centers.

Extension of the center conjecture to non-continuous centers.

Framework unifies various notions of geometric centers.

Abstract

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way to do so is to define centers as equivariant maps between $G$-spaces. In this paper, we prove that, under certain hypothesis, for any two $G$-spaces $\mathcal A,\mathcal X$, for every $V\in \mathcal A$ and for every point $P\in\mathcal X$ fixed by the symmetry group of $V$, there exists some equivariant map $\mathfrak Z:\mathcal A\to \mathcal X$ such that $\mathfrak Z(V)=P$. As a consequence of this fact, we prove an analogue (for non-neccessarily continuous centers) of the \emph{center conjecture for equifacetal simplices}, proposed by A. L. Edmonds.